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Synthesis of Driving-Point Impedances 
with Active RC Networks 

By I. W. SANDBERG 

(Manuscript received December 4, 1959) 

A general method is presented for synthesizing driving-point impedances 
using RC networks and active elements. The procedure realizes any real 
rational driving-point function and leads to rather simple structures. Only 
one active device, a negative-impedance converter, is required. The synthesis 
of biquadratic impedance functions is considered in detail, 

I. INTRODUCTION 

It is often desirable to avoid the use of magnetic elements in synthesis 
procedures, since resistors and capacitors are more nearly ideal elements 
and are usually cheaper, lighter and smaller. This is especially true in 
control systems in which, typically, exacting performance is required at 
very low frequencies. The rapid development of the transistor has pro- 
vided the network synthesist with an efficient low-cost active element 
and has stimulated considerable interest in active RC network theory 
during the past decade. '" ' 

The present paper considers the active RC synthesis of driving-pomt 
impedances. Transfer functions are not treated directly, but are covered 
at least in principle, since it is always possible, and indeed sometimes 
convenient, to reduce the synthesis of transfer functions to the synthesis 
of two-terminal impedances. 

It is now well known that any driving-point impedance function ex- 
pressible as a real rational fraction in the complex frequency variable 
can by synthesized as an active RC network requiring only one ideal 
active element. Two proofs of this result are already in the literature."'^* 
The present paper provides a third proof, although its main objective is 
to present a new and more practical realization network. The synthesis 

* Anotlier proposed proof* ia in fact concerned only with thoae impediince 
functions which are positive on some section of the negative- real axis of the 
complex frequency plane. 

947 



948 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 



technique is similar to that used in the author's first proof,^ and con- 
siderably simpler than those used in the later paper by Kinariwala.' 

Like the technique previously presented by the author,^* the present 
technique yields the values of the required networlc elements explicitly 
and directly, without the solution of simultaneous equations. The gen- 
eral procedure yields networks that contain more capacitors than the 
absolute minimum required; in return for this, however, it yields struc- 
tures which, unlike those of Kinariwala, do not need balanced amplifiers 
or complex resistance networks, f 

An alternative procedure applicable to a wide class of biquadratic 
impedance functions is also described. This procedure leads to structures 
requiring only two capacitors. 

The synthesis techniques presented in this paper are based on net- 
works employing a type of impedance converter that is a generalization 
of the negative-impedance converter. The converter concept is intro- 
duced in a general way in order to properly orient the reader. 



II. THE IDEAL IMPEDANCE CONVERTER 



Consider a two-port network terminated by an impedance Zt(s) as 
shown in Fig. 1. The input impedance at port 1 is 



ZM = hn - 



hnhiiZris) 

1 + /l22^r(s) ' 



(1) 



where the hybrid parameters are functions of the complex frequency 
variable defined by 



-El = huh + huEi , 



(2) 



II 




Iz 






+ 




+ 






E, 


Es 


Zt{s) 



















Fig. 1 — Two-port network. 



* The particular configuration involved is essentially the one mentioned in 
Section 3.2 of this paper. 

t A more practical procedure in Ref. 7 treats a restricted class of driving-point 
functions and employs a passive RC two-port network terminated by a negative 
RC impedance. Other restricted realization techniques are presented in Ref. 8. 



SYNTHESIS OF DRIVING-POINT IMPEDANCES 



949 



For the ideal impedance converter we require that for every Zt{s) 

Zi(s) - K{s)Zr{s), (3) 

where K{s) is a predetermined fixed function of the complex frequency 
variable. In order that (1) and (3) be compatible, 



(4) 



K{s) = -hi2h2i . 

An ideal impedance converter, therefore, is a two-port network with 
a hybrid parameter matrix of the form 

' /H2 

,/t21 

It follows that the impedance at port 2 with the termination Zt{s) 
connected to port 1 is 

Zt{s) 



(5) 



ZM = 



K{s)' 



(6) 



A "controlled-source" representation of the unbalanced ideal impedance 
converter is given in Fig. 2. 

The negative-impedance eonveter (huhoi ^ 1) has been heavily ex- 
ploited as a synthesis tool m active RC network theory. The synthesis 
techniques presented in this paper are based on networks employing a 
more general type of converter characterized by 



(7) 



where Z3/Z4 is the ratio of two RC driving-point unpedances. This hy- 
brid parameter matrix can be realized with either of the ideahzed cir- 



"0 


1 


z, 
z. 






-32 

-f- 



(h,2-0E; 



(ha, + Oil 



I'o 



02' 



Fig. 2 — Representation of the ideal impedance converter. 



950 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 

-o2 

1 

I 




Fig. 3 — Idealized realization of the required converter. 




Fig. 4 — Alternate idealized realization of the converter. 

cuits shown in Figs. 3 and 4.* The smaller rectangles enclose "hifinite 
gain" current amplifiers. The realization of Fig. 3 is a modification of 
Larky's idealized current-inversion negative-impedance converter. The 
larger rectangle in Fig. 4 encloses Linvill's well-known idealized voltage- 
inversion negative-impedance converter.^ Hence, (7) can be realized 
with Linvill's negative-impedance converter and two RC driving-point 
impedances. 

A controUed-source representation of (7) is given in Fig. 5. Transistor 
realizations are discussed in the Appendix. 



0(-a'' 



-3 2' 



Fig. 5 — Representation of (7). 



Many other realizations are possible. 



SYNTHESIS OF DRIVING-POINT IMPEDANCES 



951 




Fig. G — Realization one-port. 
III. SYNTHESIS OF DRIVING-POINT IMPEDANCES 

The driving-point impedance of the one-port network shown in Fig, 
6 is* 



Z(s) = 



Zi — Za 



2i 



z,- 

Z2 



(8) 



The impedance function that is to be synthesized is a real rational 
fraction in the complex frequency variable 



Z(s) = 



P(s) 



(9) 



The synthesis consists of identifying each of the four parameters Zi , 
Z2 , Z3 and Zi with a two-terminal RC impedance function. The presence 
of negative signs in (8) suggests an approach similar to techniques 
previously proposed for the synthesis of transfer functions.''^ 

Assume that the prescribed function Z(s) is positive on at least one 
section of the negative-real axis of the complex frequency plane. Sup- 
pose we write 

P(s) 



Z(s) = 



n (s + <y.) 



Qis) 



(10) 



n(s + 'T.) 



where 



* This expression can be readily obtained by applying Blackman's equation" 
to the network that results when the Ponverter is replaced by its controlled source 
representation. The required return differences are computed with respect to the 
quantity 1 + {Zs/Zt). A detailed study of Blackman's equation led to the dis- 
covery of the realization networks presented in this paper. 



952 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 

^ tri < 0-2 < 0-3 ■ ■ ■ < (Tn , 

and the number A'' is equal to the degree of the polynomial P{s) or of 
the polynomial Q(s), whichever is greater. The points — o-, are chosen 
to lie anywhere on the section or sections of the negative-real axis where 
the function Z{s) is finite and positive.* 

We replace both the numerator and denominator of the right-hand 
side of (10) by their partial fraction expansions and group the resulting 
terms to obtain 



Z{s) = ^ ^^ (U) 



Pi 


P2 


Qi 


Q2 


Ps 


Pa 


Qi 


Q2 



where 

N 

QiQi = n (5 + cTi) 

and Pi/Qi , P2/Q2 , PJQi , and P4/Q2 are each RC driving-point imped- 
ances. The function Z{s) is expressible in this form since: 

(a) Each of the two partial fraction expansions possesses real residues 
of like sign at any particular pole. 

(b) All RC driving-point impedances are expressible in the form 



Rca ~\~ Aj 



ttk 



k=i s -\- bk' 



where R^ , ak and 6t are nonnegative. 
Equation (11) is equivalent to 



Pi P 



^^'^ - P3 + W1 pI-Trq. - ^^^^ 

We identify the impedance parameters of (8) as follows: 

7 _ Pi 7 _ P2 

'^' '^' (13) 

7 ^ Pi 7 ^ ^2 

^ P^^RQ^ ' Pa^RQi' 

* The choice of the ff.'s influences the spread of element values and the sensi- 
tivity of the driving-point impedance to variations in the active and passive 
elements. 



SYNTHESIS OF DRIVIXG-POINT IMPEDANCES 953 

According to the qualifications appended to (11), the functions Z3 and 
Z^ are RC driving-point impedances. 

Consider the function Zi . The degree of the polynomial Qi is at least 
as great as the degree of P3 . Note that, as the parameter R assumes 
nonnegative real values that increase from zero to infinity, the poles of 
Zi move from the zeros of P3 to the zeros of Qi . Recall that the ratio of 
polynomials Pi/Qi is an RC driving-point impedance. Consequently, it 
is always possible to find a finite value of the parameter R for which the 
poles and zeros of Zi interlace properly. 

A similar argument shows that Z2 also can always be made an RC 
driving-point impedance. Note that Z(s) need not be a positive-real 
function. 

3.1 -471 Example 
Let 

Z(s)=i±^. (14) 

This function is positive on the entire negative-real axis. We choose 
0-1 = 1, and (72 = 2. From (12), 

s + 2 _ 3 

7/ ■, s + 1 s + 2 /,.N 

^^ s(l + fl) + 3 + /2 Rs+±±2R- ' 



s + 1 s+ 2 



Employing (13), 



^4 - 1 + ^-T , Z,= '^ 



s + 1 ' ' s + 2' 



(16) 



s(l + /?) + 3 + P' - /2s + 4 + 2/2' 

The sot of unpedances is realizable for R ^ \. Choose 72 ^ 1 so that 

The corresponding network is shown in Fig. 7. 

Note that Zz and Z4 can be scaled by the same constant without af- 
fecting the impedance Z{s). In a practical design, this degree of freedom 
would be utilized in order to optimize some figure of merit such as the 
sensitivity function. The choice of the tr.'s would be similarly influenced. 



954 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 19G0 




Fig. 7 — Realization of (14). 

3.2 An Alternate Realization 

The class of functions treated in this section can also be synthesized 
with the network given in Fig. 8. The input impedance of this structure 



Z(.s) = 






(18) 



By employing arguments similar to those already discussed, it can be 
shown that the value of a nonnegative real parameter R can be chosen 
to ensure that the following set of impedances is realizable:* 



Zs = 
Z, - 



P, + RQi 



Pi 

P3 + fiQl' 



Z, = 

z, = 



p, + RQ2 
Q2 ' 

Pi 

Pa + RQ2' 



(19) 




Fig. 8 — Alternate realization one-port. 



* A realizable set of impedaacea can also be obtained in other ways. 



SYNTHESIS OF DRIVING-POINT IMPEDANCES 955 

This circuit has the disadvantage of requiring a "floating" two-port 
active network. 

3.3 Reslnclions on Z{s) 

The methods of synthesis presented in the previous sections are not 
appHcable to functions that are nonpositive on the entire negative-real 
axis. This is a significant theoretical restriction, since positive-real func- 
tions, for example, need not possess the requu-ed property. In particular, 
all reactance functions must be excluded. 

The difficulty mentioned above can be circumvented m several ways 
by modifications of the synthesis technique. Suppose that the prescribed 
impedance Z{s) is nonpositive on the entire negative-real axis. The 
function 

Z'{s) = Z(s) - — ^ ao > 0, b, ^ (20) 

s -\- bo 

must, however, be positive on one section of the negative-real axis. It can 
therefore be synthesized by the previously discussed procedure. The 
impedance Z(s) is obtained by connecting an RC impedance a^/is 4- bo) 
in series with the resulting network. An alternative procedure on an 
admittance basis also applies, the network being modified at the input 
terminals by the parallel connection of an RC impedance 

Co + - do>0, Co ^ 0. 
s 

Both methods usually necessitate a larger number of passive com- 
ponents than would be required for the synthesis of —Z(s). For this 
reason it may be more desirable to employ a negative-impedanc^e con- 
verter terminated by —Z(s). 

3.4 Sufficiency of One Active Element or One Negative-Impedance Converter 

Since the realization of the converter requires only one active element 
or only one Linviil-type negative-impedance converter, the preceding 
discussion constitutes a proof of 

Theorem: Ani/ driving-point impedance function, expressible as a real 
rational fraction in the complex frequency variable, can be synthesized with 
a network containing only resistors, capacitors and either a single ideal ac- 
tive element or a single ideal negative-impedance converter. 

Note that it is theoretically possible to synthesize impedance func- 
tions which approach infuiity as any integral power of the frequency 



956 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 

variable. Obviously, these functions can be realized by actual networks 
only over the frequency band where the active two-port is essentially 
characterized by the controlled-source model of Fig. 5.* 

IV. BIQUADRATIC SYNTHESIS 

The synthesis of biquadratic impedance functions merits special at- 
tention. The network associated with a function of this type, previously 
considered as an example, requires a total of four capacitors (two in the 

converter). Two of the capacitors can be eliminated by employing an 
alternative technique based on the network of Fig. 6 with Zi and Zi re- 
placed by resistors R\ and Ri . The structure becomes an impedance 
converter imbedded in a simple resistance network. 
From (8), 

Zs _ RiJZ — Ri) .^ . 

Z4 Ri(Z — R2) 

The biquadratic function Z(s) is given by 

^ ( .-.0(s-..) 

(s~ pi)(s -P2) 

It is required that Ri and R2 be chosen so that Z3 and Z4 are RC imped- 
ances. Assume that each of the impedances Z3 and Zt is to be realizable 
as a resistor in series with a parallel combination of a resistor and capaci- 
tor as sliown in Fig. 9. This structure is sufficient to realize the most 
general first-degree RC impedance function. The ratio of these two func- 
tions is given by 

Zi (s 4- asJCs + ai) 



where, from (21) and (22), 



''-wr^y <^) 




Fig. 9 — Structural form of Z3 or Zt . 



• It must be remembered that this model will ordinarily be inadequate for 
stability analyses. For this purpose it must be modified to be valid in the fre- 
quency range where the significant active and passive parasitic parameters are 
influential. 



SYNTHESIS OF DKIVING-POINT IMPEDANCES 



957 



Four possibilities exist for the pole-zero pattern of 23/Z4 as shown in 
Fig. 10. The zeros in Fig. 10(a) and the poles in Fig. 10(b) may occur 
with multiphcity two. 

Assume tentatively that Z(s) has complex conjugate poles and zeros. 
The function Z(a)\- « ^ a ^ ^], where s = a -\- jui, is nonnegative 
and approaches unity at both extremes of the argument. Since the func- 
tion is the ratio of two second-degree polyi^omials ui a, only two points 



J*^ 



(a) — ^ — e — e — ^ 

(b) — e — H — ^ — e- 

(c) — e — X — e — K- 

(d) — X — e — X — e- 



-tr 



Fig. 10 — Permissible pole-zero patterns for Zz/Zt . 

of intersection with a horizontal hne are possible. When such an inter- 
section occurs, the intersecting points will be separated by an cxtrcmum 
of the function. Hence, if Z{s') exhibits at least one extremum on the 
negative-real axis, the parameters Ri and R2 can be chosen to provide a 
pole-zero pattern for Z3/Z4 of the type shown in Figs. 10(a) or (b). 
Since, for both type (a) and type (b), (1 - Ri) and (1 - Sa) have the 
same sign, the impedances Z3 and Z4 would be realizable. 





Fig. 11 — Construction of the extremum points of Zia). 

Consider the pole-zero diagram for Z(s). It can easily be shown that 
the circle passing through the poles and zeros [Fig. 11(a)] will intersect 
the real axis at the two pomts w^here Z((j) has an extremum. Conse- 
quently, any open-circuit or short-circuit stable biquadratic impedance 
function with complex conjugate poles and zeros can be synthesized. 
Two permissible limithig cases exist when the zeros or poles or both the 
zeros and poles occur with multiplicity two on the real axis. Another 



958 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1 9(l0 

permissible case occurs when the zeros are at infinity and the complex 
conjugate poles are in the left-half plane. 

Suppose that Z(s) has distinct real zeros and complex conjugate 
poles. It can easily be shown that the circle centered on the real axis at 

'^o = ^T 7^> \ Pi-2 = '^p ^J^p (25) 

and passing through the poles [Fig. 11(b)] will intersect the real axis at 
the two points where Z(a) has an extremum. The circle must pass be- 
tween the zeros. It follows from (25) that the center of the circle will lie 
to the left of the zeros if the poles lie to the left of the point midway 
between the zeros. This will result m an extremum of Z(o-) located to 
the left of the poles in a region where the function is positive. It follows 
directly that the synthesis can be accomplished for any biquadratic 
impedance function with left-half plane complex conjugate poles and 
real zeros, where the poles are located to the left of the point midway 
between the zeros.* 

The steps in the biquadratic synthesis procedure are: (a) choose Ri 
and Rz so that Zs and Z4 are reahzable, and (b) from (21), identify Z3 
and Zi . The permissible values of Ri and R2 can be determined by 
inspection of Z{a). 

It should be noted that this procedure is not limited to the two broad 
classes of functions considered above. The synthesis can obviously be 
accomplished if step (a) can be carried out. Hence the applicability of 
the procedure can be determined by inspection of Z((t). 

4.1 An Example 
Let 

Z(s)= '\-J'^^ . (26) 

It is evident from the graph of Z(a) for this case (Fig. 12) that the 
choice Ri = A, R2 = G is acceptable. From (21) and (26), 

Z3 QZ ~ 4 9 s^ -\-is-\-i 



(27) 



Z4 4Z - 6 10s^+ ?s+ i 

_ 9 (s+i)(s + 2) 
10 (s + i)(s+ 1)- 

Two pos.sibilities exist for the pair of impedances Z3 and Z4 : 

* Biquadratic admittance functions with distinct negative-real polea can be 
realizea with a negative-impedance converter as the difference of two RC ad- 
mittances, For this reason, such functions are not considered in detail here. 



SYNTHESIS OF DRIVING-POINT IMPEDANCES 



959 



Z, = ^ 



or 



9 (s + 2) 

10 (s + 1) ' 



Z^ =^ 



9 (s + 2) 



10 (s + I) ' 



Z4 = 



Z4 = 



(s + ^) 



(^ + 1) 

(s + i) ' 



(28) 



(29) 



4.2 Si/nthesis on the Basis of the Network Shown in Fig. 8 

The synthesis technique presented in this section can be extended to 
apply to the network of Fig. 8 by comparing (8) and (18) and identify- 
ing 



7 -^'7 



Zg — Z4 , 

Z^ = Ri . 



(30) 



V. CONCLUSION 

A general method of synthesizing driving-point impedances has been 
presented. An impedance converter is required that can be reahzed by 
modifying Larky's current-Inversion negative-impedance converter. An 
alternate realization employs Tjinvill's voltage-inversion negative-im- 
pedance converter and two RC impedances. This realization leads to 
the result that any driving-point impedance function, expressible as a 
real rational fraction in the complex frequency variable, can be syn- 
thesized as a network containing resistors, capacitors and a single nega- 



2(<r) 



























} 


\ 










/ 














/ 










^ 


^ 


^ 






















V 
















\ 





-6 -5 -4 -3 -2 -1 



Fig. 12 — Graph of Z{a) for a biquadratic function. 



960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 

tive-impedance converter. The technique does not require the synthesis 
of two-port RC or ?i-port balanced resistor networks, and leads to the 
direct determination of the required two -terminal RC elements. 

The synthesis of biquadratic impedance functions has been given 
special attention^ resulting in structures employing the minimum number 
of capacitors and a moderate number of resistors. The procedure is 
applicable to a wide class of functions including, m particular, all open- 
circuit or short-circuit stable impedances with complex conjugate poles 
and zeros. 

VI. ACKNOWLEDGMENT 

The author is grateful to B. McMillan and to P. M. Dollard for their 
constructive criticism and advice. 

APPENDIX 

Transistor Circuit Analysis 

An approximate model for the transistor in the range where its pa- 
rameters are essentially independent of frequency is shown in Fig. 13 

ale 

(e) l^ VA » O = (c) 



Fig. 13 — Approximate low-frequency transistor model. 

The collector resistance is assumed to be infinite, an approximation 
whichis often reasonable — especially fordrift transistors. In terms of this 
model, the hybrid parameters of the impedance converter circuits of 
Fig. 14 are 



Fig. 14(a): 



h = rid - ai)(Zz-\-Z,) 

" Z,(0,ay -f- 1) + Z3 + (1 - „,)ri -''' 



hvi - 1, ;i22 = 0, 

Zz_ (1 - ai) 

I, Z4 ai^2 \ Zi / Zz 

fhi — — 



0+^0 



(31) 



1+^ 



1 4- ^3 + ^1^1 ~ «l) 1 ^* ' 



1o- 



SYNTHESI8 OF DRIVING-POINT IMPEDANCES 

T, 



961 




T, 



-oZ 1 o- 




Ui 




-02 1 c- 




Z4 -< 



Ca) 



(b) 



\'o- 



Z3 Z4 



-o2 1 o- 



Z3 Z4 



CO (d) 

Fig. 14 — Transistor realizations of the converter. 



wliere 



r = r. + ^^ /3= " 



Fig. 14(b): 



hn = 



1 — a 1 — a 

-n(l - ax)(Zz + Z,) 
a,&,Z,- (1 - ai)(ri + Z3+ Z^) 

hi2 = 1 , h.2i - , 

2^3 + rid - 



Z3 j_ r Zs+rid - ai) "| 



_ d - aO [, 1 n+Z, 1 

Fig. 14(c): 

/til = (1 - ai)n ^0, 
^1.12 ^ 1, '122 = 0, 



0, 



Z4* 



■02 



-02 



(32) 



9G2 THE BELL SYSTEM TECHNICAL JOUHNAL, JULY 1960 

(33) 



All = 



0(ia2 ^ — (1 — ai) 1 + (1 — aa) -^-- — -^ 
1 + (1 - 0:2; — 7? — 



Fig. 14(d): 

J, ^ _/, _ . Z3+ (1 - a2)(Z4 + r2) 

^' "'^'^ a,a,Z, - {1 - aOl^a + (1 " "2) (Z4 + r.)] ^ ' 

/112 ^ 1, fhz = 0, 



/t>;.i ^ 



l^^^-->0^l) 



l3 ,_ ,, „.^A ^^A (34) 

Za 



mai 



- (1 - «.) 



Za + (1 - a2)(r2 + Z 



Z4 



^£}1 ^4- 



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1. Dietzold, R. L., U. S. Patent 2,5i9,065, April 17, 1951. 

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1945, p. 329. 

3. Linvill, J. G., RC Active Filters, Proc. I. RE., 42, March 1954, p. 555. 

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5. Yanagisawii, T., RC Active Networks Using Current- Inversion-Type Nega- 

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